Mathematical Logic for Computer Science

This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere in much more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to sel

Now available in paperback, this celebrated book has been prepared with readers' needs in mind, remaining a systematic guide to a large part of the modern theory of Probability, whilst retaining its vitality. The authors' aim is to present the subject of Brownian motion not as a dry part of mathematical analysis, but to convey its real meaning and fascination. The opening, heuristic chapter does just this, and it is followed by a comprehensive and self-contained account of the foundations of theory of stochastic processes. Chapter 3 is a lively and readable account of the theory of Markov processes. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

Oriented matroids are a very natural mathematical concept which presents itself in many different guises and which has connections and applications to many different areas. These include discrete and computational geometry, combinatorics, convexity, topology, algebraic geometry, operations research, computer science and theoretical chemistry. This is the second edition of the first comprehensive, accessible account of the subject. It is intended for a diverse audience: graduate students who wish to learn the subject from scratch; researchers in the various fields of application who want to concentrate on certain aspects of the theory; specialists who need a thorough reference work; and others at academic points in between. A list of exercises and open problems ends each chapter. For the second edition, the authors have expanded the bibliography greatly to ensure that it remains comprehensive and up-to-date, and they have also added an appendix surveying research since the work was firs

Self-organized criticality (SOC) is based upon the idea that complex behavior can develop spontaneously in certain multi-body systems whose dynamics vary abruptly. This book is a clear and concise introduction to the field of self-organized criticality, and contains an overview of the main research results. The author begins with an examination of what is meant by SOC, and the systems in which it can occur. He then presents and analyzes computer models to describe a number of systems, and he explains the different mathematical formalisms developed to understand SOC. The final chapter assesses the impact of this field of study, and highlights some key areas of new research. The author assumes no previous knowledge of the field, and the book contains several exercises. It will be ideal as a textbook for graduate students taking physics, engineering, or mathematical biology courses in nonlinear science or complexity.

Computational learning theory is a subject which has been advancing rapidly in the last few years. The authors concentrate on the probably approximately correct model of learning, and gradually develop the ideas of efficiency considerations. Finally, applications of the theory to artificial neural networks are considered. Many exercises are included throughout, and the list of references is extensive. This volume is relatively self contained as the necessary background material from logic, probability and complexity theory is included. It will therefore form an introduction to the theory of computational learning, suitable for a broad spectrum of graduate students from theoretical computer science and mathematics.

This book is about abduction, 'the logic of Sherlock Holmes', and about how some kinds of abductive reasoning can be programmed in a computer. The work brings together Artificial Intelligence and philosophy of science and is rich with implications for other areas such as, psychology, medical informatics, and linguistics. It also has subtle implications for evidence evaluation in areas such as accident investigation, confirmation of scientific theories, law, diagnosis, and financial auditing. The book is about certainty and the logico-computational foundations of knowledge; it is about inference in perception, reasoning strategies, and building expert systems.

This volume contains survey papers by the invited speakers at the Conference on Semigroup Theory and Its Applications which took place at Tulane University in April, 1994. The authors represent the leading areas of research in semigroup theory and its applications, both to other areas of mathematics and to areas outside mathematics. Included are papers by Gordon Preston surveying Clifford's work on Clifford semigroups and by John Rhodes tracing the influence of Clifford's work on current semigroup theory. Notable among the areas of application are the paper by Jean-Eric Pin on applications of other areas of mathematics to semigroup theory and the paper by the editors on an application of semigroup theory to theoretical computer science and mathematical logic. All workers in semigroup theory will find this volume invaluable.

The book covers elementary aspects of category theory and topos theory for graduate students in mathematics, computer science, and logic; it has few mathematical prerequisites, and uses categorical me

The fundamental ideas concerning computation and recursion naturally find their place at the interface between logic and theoretical computer science. The contributions in this book, by leaders in the field, provide a picture of current ideas and methods in the ongoing investigations into the pure mathematical foundations of computability theory. The topics range over computable functions, enumerable sets, degree structures, complexity, subrecursiveness, domains and inductive inference. A number of the articles contain introductory and background material which it is hoped will make this volume an invaluable resource.

This book offers a new, algebraic, approach to set theory. The authors introduce a particular kind of algebra, the Zermelo-Fraenkel algebras, which arise from the familiar axioms of Zermelo-Fraenkel set theory. Furthermore the authors explicitly construct such algebras using the theory of bisimulations. Their approach is completely constructive, and contains both intuitionistic set theory and topos theory. In particular it provides a uniform description of various constructions of the cumulative hierarchy of sets in forcing models, sheaf models and realisability models. Graduate students and researchers in mathematical logic, category theory and computer science should find this book of great interest, and it should be accessible to anyone with some background in categorical logic.

This book is about abduction, 'the logic of Sherlock Holmes', and about how some kinds of abductive reasoning can be programmed in a computer. The work brings together Artificial Intelligence and philosophy of science and is rich with implications for other areas such as, psychology, medical informatics, and linguistics. It also has subtle implications for evidence evaluation in areas such as accident investigation, confirmation of scientific theories, law, diagnosis, and financial auditing. The book is about certainty and the logico-computational foundations of knowledge; it is about inference in perception, reasoning strategies, and building expert systems.

The concepts of a locally presentable category and an accessible category have turned out to be useful in formulating connections between universal algebra, model theory, logic and computer science. The aim of this book is to provide an exposition of both the theory and the applications of these categories at a level accessible to graduate students. Firstly the properties of l-presentable objects, locally l-presentable categories, and l-accessible categories are discussed in detail, and the equivalence of accessible and sketchable categories is proved. The authors go on to study categories of algebras and prove that Freyd's essentially algebraic categories are precisely the locally presentable categories. In the final chapters they treat some topics in model theory and some set theoretical aspects. For researchers in category theory, algebra, computer science, and model theory, this book will be a necessary purchase.

This 1994 book develops a way of representing the meanings of linguistic expressions which is independent of any particular language, allowing the expressions to be manipulated in accordance with rules related to their meanings which could be implemented on a computer. It begins with a survey of the contributions of linguistics, logic and computer science to the problem of representation, linking each with a particular type of formal grammar. A system of graphs is then presented, organized by scope relations in which linguistic constituents are sub-graphs whose configuration is determined by their categories. In developing this system, the author extends the notion of scope and argues that anaphoric and relative pronouns are structural signs not linguistic constituents. Certain count nouns are made the basis of this system and an account of proper names relating the count nouns, is given.

This is an up-to-date and integrated introduction to model theory, designed to be used for graduate courses (for students who are familiar with first-order logic), and as a reference for more experienced logicians and mathematicians. Model theory is concerned with the notions of definition, interpretation and structure in a very general setting, and is applied to a wide variety of other areas such as set theory, geometry, algebra (in particular group theory), and computer science (e.g. logic programming and specification). Professor Hodges emphasises definability and methods of construction, and introduces the reader to advanced topics such as stability. He also provides the reader with much historical information and a full bibliography, enhancing the book's use as a reference.

This 1991 volume contains the proceedings of the first international workshop on Logical Frameworks. The contributions are concerned with the application of logical reasoning and proof theory in computer science and its relevance to automatic theorem proving, and consequently topics such as artificial intelligence. It is the only source for much of this material and will be a necessary purchase for mathematicians and computer scientists undertaking research at the interface of logic and software engineering.

This textbook is an introduction to denotational semantics and its applications to programming languages. Dr Allison emphasizes a practical approach and the student is encouraged to write and test denotational definitions. The first section is devoted to the mathematical foundations of the subject and sufficient detail is given to illustrate the fundamental problems. The remainder of the book covers the use of denotational semantics to describe sequential programming languages such as Algol, Pascal and C. Throughout, numerous exercises, usually in Pascal, will help the student practise writing definitions and carry out simple applications. The book culminates in discussing an executable semantics of the logic-programming language Prolog. Being an introduction, advanced undergraduates in computer science and graduates new to the subject will find this a readily accessible account of one of the central topics of computer science.

In this book, which was originally published in 1985, Arto Salomaa gives an introduction to certain mathematical topics central to theoretical computer science: computability and recursive functions, formal languages and automata, computational complexity and cryptography. Without sacrificing readability, the presentation is essentially self-contained, with detailed proofs of all statements provided. Professor Salomaa is well known for his books in this area. The present work provides an insight into the basics, together with explanations of some of the more important developments in the field.

Recursion theory – now a well-established branch of pure mathematics, having grown rapidly over the last 35 years – deals with the general (abstract) theory of those operations which we conceive as being `computable' by idealized machines. The theory grew out of, and is usually still regarded, as a branch of mathematical logic. This book is a collection of advanced research/survey papers by eminent research workers in the field, based on their lectures given at the Leeds Logic Colloquium 1979. As such it provides an up-to-date view of current ideas and developments in the field of recursion theory as a whole. The individual contributions fit together naturally so as to provide an overview of all the main areas of research in the field. It will therefore be an important and invaluable source for advanced researchers and research students in mathematics and computer science (particularly in Europe, USA and USSR).

By the time students have done some programming in one or two languages and have learnt the common ways of representing information in a computer, they will want to embark upon further study of theoretical or applied topics in computer science. Most will encounter problems that require for their solution one or more of the techniques described in this book: for example problems depending upon the formation and solution of different equations; the task of making lists of possible alternatives and of answering questions about them; or the search for discrete optima. Written by the same authors as the highly successful Information Representation and Manipulation in a Computer, this book describes algorithms of mathematical methods and illustrates their application with examples. The mathematical background needed is elementary algebra and calculus. Numerous exercises are provided, with hints to their solutions.

Introduction to Logic Design by Alan Marcovitz is intended for the first course in logic design, taken by computer science, computer engineering, and electrical engineering students. As with the prev

The thirteen chapters written expressly for this book by logicians, theoretical computer scientists, philosophers, and semanticists address, from the perspective of mathematical logic, the problems of

Constraint logic programming lies at the intersection of logic programming, optimisation and artificial intelligence. It has proved a successful tool in many areas including production planning, transportation scheduling, numerical analysis and bioinformatics. Eclipse is one of the leading software systems that realise its underlying methodology. Eclipse is exploited commercially by Cisco, and is freely available and used for teaching and research in over 500 universities. This book has a two-fold purpose. It's an introduction to constraint programming, appropriate for one-semester courses for upper undergraduate or graduate students in computer science or for programmers wishing to master the practical aspects of constraint programming. By the end of the book, the reader will be able to understand and write constraint programs that solve complex problems. Second, it provides a systematic introduction to the Eclipse system through carefully-chosen examples that guide the reader throu

This is the second volume of a unique collection that brings together the best English-language problems created for students competing in the Computational Linguistics Olympiad. These problems are representative of the diverse areas presented in the competition and designed with three principles in mind:· To challenge the student analytically, without requiring any explicit knowledge or experience in linguistics or computer science;· To expose the student to the different kinds of reasoning required when encountering a new phenomenon in a language, both as a theoretical topic and as an applied problem;· To foster the natural curiosity students have about the workings of their own language, as well as to introduce them to the beauty and structure of other languages;· To learn about the models and techniques used by computers to understand human language.Aside from being a fun intellectual challenge, the Olympiad mimics the skills used by researchers and scholars in the field of computa

This is the first volume of a unique collection that brings together the best English-language problems created for students competing in the Computational Linguistics Olympiad. These problems are representative of the diverse areas presented in the competition and designed with three principles in mind:· To challenge the student analytically, without requiring any explicit knowledge or experience in linguistics or computer science;· To expose the student to the different kinds of reasoning required when encountering a new phenomenon in a language, both as a theoretical topic and as an applied problem;· To foster the natural curiosity students have about the workings of their own language, as well as to introduce them to the beauty and structure of other languages;· To learn about the models and techniques used by computers to understand human language. Aside from being a fun intellectual challenge, the Olympiad mimics the skills used by researchers and scholars in the field of computa

In Human Reasoning and Cognitive Science, Keith Stenning andMichiel van Lambalgen--a cognitive scientist and a logician--argue for the indispensability ofmodern mathematical logic to the study of huma

In Human Reasoning and Cognitive Science, Keith Stenning and Michiel van Lambalgen—a cognitive scientist and a logician—argue for the indispensability of modern mathematical logic to the

"The ability to reason and think in a logical manner forms the basis of learning for most mathematics, computer science, philosophy and logic students. Based on the author's teaching notes at the Univ

Rosen's Discrete Mathematics and its Applications presents a precise, relevant, comprehensive approach to mathematical concepts. This world-renowned best-selling text was written to accommodate the needs across a variety of majors and departments, including mathematics, computer science, and engineering. As the market leader, the book is highly flexible, comprehensive and a proven pedagogical teaching tool for instructors.

In the past several decades the classical Perron–Frobenius theory for nonnegative matrices has been extended to obtain remarkably precise and beautiful results for classes of nonlinear maps. This nonlinear Perron–Frobenius theory has found significant uses in computer science, mathematical biology, game theory and the study of dynamical systems. This is the first comprehensive and unified introduction to nonlinear Perron–Frobenius theory suitable for graduate students and researchers entering the field for the first time. It acquaints the reader with recent developments and provides a guide to challenging open problems. To enhance accessibility, the focus is on finite dimensional nonlinear Perron–Frobenius theory, but pointers are provided to infinite dimensional results. Prerequisites are little more than basic real analysis and topology.

The father of modern computer science, Alan Turing paves the way for esoteric mathematical computations that, when used by Nazi Germany's Ahnenerbe-SS to perform a summoning, results in an unexpected

This text is intended primarily for readers interested in mathematical probability as applied to mathematics, statistics, operations research, engineering, and computer science. It is also appropriate

George Boole, for whom Boolean logic is named, developed the idea of expressing any idea as a mathematical or logical statement. Today, Boolean logic is foundational to computer programs and computer

This monograph provides an intensive course for graduate students in computer science, as well as others interested in extensions of logic programming, on the theoretical foundations of disjunctive lo